Billy Jones and David Noble struck sensational free-kicks to earn Exeter the win, their first success at home since the end of November.
Wednesday opened the scoring in the 65th minute when a long throw-in was cleared as far as Jermaine Johnson and he lashed the ball into the bottom corner from 25 yards.
But Exeter hit straight back as Jones curled a superb free-kick round the Wednesday wall and into the bottom corner after a foul on Daniel Nardiello.
With six minutes remaining, the Grecians went in front through David Noble's stunning free-kick. The ball seemed to be heading way over the crossbar, but it dipped and hit the upright before nestling in the back of the net.
The result leaves Wednesday seven points behind Charlton, who saw their game against Chesterfield called off as the League One programme was reduced to just three games due to adverse weather conditions.
Will Hoskins' debut goal helped Sheffield United to a 3-0 win over Wycombe as Danny Wilson's side moved up to third place in League One.
Substitute Hoskins pounced in the 66th minute, five minutes after replacing Richard Creswell, before Ched Evans and Kevin McDonald sealed the points to leave the Blades just one behind Sheffield Wednesday with two games in hand over their city rivals.
Elsewhere, the game between Hartlepool and Bournemouth ended in a 0-0 draw.